Shuai Ma

Xiaoteng Ma, Shuai Ma, Li Xia et al. · tsinghua

Keeping risk under control is often more crucial than maximizing expected rewards in real-world decision-making situations, such as finance, robotics, autonomous driving, etc. The most natural choice of risk measures is variance, which penalizes the upside volatility as much as the downside part. Instead, the (downside) semivariance, which captures the negative deviation of a random variable under its mean, is more suitable for risk-averse proposes. This paper aims at optimizing the mean-semivariance (MSV) criterion in reinforcement learning w.r.t. steady reward distribution. Since semivariance is time-inconsistent and does not satisfy the standard Bellman equation, the traditional dynamic programming methods are inapplicable to MSV problems directly. To tackle this challenge, we resort to Perturbation Analysis (PA) theory and establish the performance difference formula for MSV. We reveal that the MSV problem can be solved by iteratively solving a sequence of RL problems with a policy-dependent reward function. Further, we propose two on-policy algorithms based on the policy gradient theory and the trust region method. Finally, we conduct diverse experiments from simple bandit problems to continuous control tasks in MuJoCo, which demonstrate the effectiveness of our proposed methods.

Shuai Ma, Guangwu Liu, Li Xia

Sharpe ratio (also known as reward-to-variability ratio) is a widely-used metric in finance, which measures the additional return at the cost of per unit of increased risk (standard deviation of return). However, the optimization of Sharpe ratio in Markov decision processes (MDPs) is challenging, because there exist two difficulties hindering the application of dynamic programming. One is that dynamic programming does not work for fractional objectives, and the other is that dynamic programming is invalid for risk metrics. In this paper, we study the Sharpe ratio optimization in infinite-horizon MDPs, considering both the long-run average and discounted settings. We address the first challenge with the Dinkelbachs transform, which converts the Sharpe ratio objective to a mean-squared-variance (M2V) objective. It is shown that the M2V optimization and the original Sharpe ratio optimization share the same optimal policy when the risk-sensitive parameter is equal to the optimal Sharpe ratio. For the second challenge, we develop an iterative algorithm to solve the M2V optimization which is similar to a mean-variance optimization in MDPs. We iteratively solve the M2V problem and obtain the associated Sharpe ratio that is used to update the risk-sensitive parameter in the next iteration of M2V problems. We show that such a sequence of Sharpe ratios derived is monotonically increasing and converges to the optimal Sharpe ratio. For both average and discounted MDP settings, we develop a policy iteration procedure and prove its convergence to the optimum. Numerical experiments are conducted for validation. To the best of our knowledge, our approach is the first that solves the Sharpe ratio optimization in MDPs with dynamic programming type algorithms. We believe that the proposed algorithm can shed light on solving MDPs with other fractional objectives.